which can be solved through a Singular Value Decomposition (SVD). However, for a PDE that evolves in time such as the convection-diffusion equations for dye simulations, time integration can incur significant computational expense (where the SVDs must be applied at each timestep in multiple spatial dimensions). For tractability, this work employs an accelerated continuation technique known as FC(Gram) [1]}, [2]}, [3]}. Relying on the use of very small (e.g., 5) numbers of function values near the left (\(x=0\) ) and right (\(x=1\) ) endpoints, the FC(Gram) algorithm projects such values onto an interpolating (Gram) polynomial basis. Fourier continuations are precomputed for each (significantly reduced) basis vector through solving the corresponding least squares formulation similar to Equation (REF ) by a high-precision or symbolic SVD. That is, via a subset of given function values on small numbers \(d_\ell \)  and \(d_r\)  of matching points
\(\lbrace x_0,..,x_{d_\ell -1}\rbrace \)  and \(\lbrace x_{N-d_r},...,x_{N-1}\rbrace \)  at the leftmost and rightmost sub-intervals of \([0,1]\) , one can produce the discrete periodic extension of size \(N_\text{cont}\)  by projecting these end values onto a Gram polynomial basis up to degree \(d_\ell -1\)  (or \(d_r-1\) ). This constructs an interpolating polynomial whose Fourier continuations can be precomputed, where orthogonality is enforced by the scalar product naturally defined by the discretization points. This ultimately forms a (precomputed) “basis" of continued functions that can represent any function \(c\)  and provide a smooth transition from \(c(x=0)\)  to \(c(x=1)\)  over the slightly larger interval \([0,b]\) .
